

\documentstyle[11pt,psfig]{article}
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\title{ A simple introduction  of  LiNbO3 optical superlattice}
\author{            Haifeng Tu\\
                   Department of Physics\\
                 University of Cincinnati \\
                   Cincinnati,Ohio 45221    \\
       } 

\begin{document}
\maketitle           
 \begin{abstract}
 $ LiNbO_{3} $   optical superlattice prepared by doped Czochralski
method has great potential applications in optical storage and laser
printing. Using quasiphase matching
(QPM), the frequency upconversion can be realized in the violet-blue
range. The largest nonlinear optical coefficient $ d_{33} $  is
 utilized, the  great enhancement of output power is achieved. This
 paper is a simple introduction of this important achievement in the
nonlinear optical materials field. \end{abstract}

 \section{The Second Harmonic Generation  and the 
    Quasiphase Matching theory}

  Compact and reliable short wavelength light sources have potential
applications in optical storage and laser printing. The output of such
sources is required to be as short as possible in the visible region due
to the fact that the optical storage density is in inverse
proportion to the square of the writing wavelength. Frequency conversion
of laser diodes
using nonlinear optical interaction in crystals is a valuable technology
for obtaining light
in those spectral regions which are difficult to attain with laser
diodes.

   Second harmonic generation (SHG) by quasiphase matching (QPM) in a
$ LiNbO_{3} $
optical superlattice crystals and waveguides has recently attracted a
great deal of attention. By modulating the optical susceptibility with
periodic ferroelectric domain
structures which have an appropriate period (i.e., $ LiNbO_{3} $
optical superlattice with a period in the range of several microns),it is
possible to realize quasiphase match at an
arbitrary temperature over the crystal’s entire transparency range. To
get efficient
conversion ,the quasiphase matching condition has to be fulfilled,which
requires pump
sources of specific wavelength and a careful choice of modulation period
in the optical
superlattice.

    The conditions required for efficient energy conversion in a second
harmonic generation is energy conservation
    \begin{equation}
     2\omega_{f}=\omega_{h}
    \end{equation}
and momentum conservation
  \begin{equation}
   2K_{f}-K_{h}=\Delta K
  \end{equation}
where $\omega_{f} $  and  $\omega_{h} $  represent the angular frequencies of
fundamental and second harmonic
light and  $ K_{i} (i=f,h) $    are wave vectors for the interacting 
waves. $\Delta K $  represents the phase mismatch. For wavelengths where the
momentum conservation can not  be achieved (i.e., $ \Delta K \neq 0 $ ),
a grating maybe used to compensate for the phase mismatch.

   The grating period $ \Gamma$ should be chosen to be
\begin{equation}
\Gamma=m\frac{ 2\pi}{\Delta K}
\end{equation}
 where $ m $ is an integer that gives the order of the phase matching
process.
    And $\Gamma=2Lc$, and $Lc$   is the coherent length of the crystal, given
by
 \begin{equation}
 Lc=\frac{\lambda}{4(n_{2}-n_{1})}
 \end{equation}
Where $\lambda$ is the fundamental wavelength, and $n_{1}, n_{2}$
  are the
refractive indices at the fundamental and second-harmonic wavelengths,
respectively.
In SHG QPM, the maximum power that can be generated at  $\omega_{h}$
  is
\begin{equation}
P(\omega_{h})=CP(\omega_{f})^{2}/m^{2},  P(\omega_{h})\propto
 P(\omega_{f})^{2}
\end{equation}
Where $C$ is a constant composed of several terms one of which is $
d_{xx}^{2}$  . $d_{xx} $  is the nonlinear optical coefficient and
$P(\omega_{i}) $  the power at $\omega_{i} (i=f,h) $.

\section{$ LiNbO_{3} $ optical superlattice}
 $LiNbO3$    is a commonly used nonlinear optical crystal; however, its 
largest nonlinear
coefficient $ d_{33} $   is not phase matchable, so it has not  been utilized
at all. Since $d_{33} $  is
about 7.5 times larger  than  $d_{31} $ ,if we fabricated a $LiNbO_{3}$
crystal with periodic laminar
ferroelectric domains,alternative in polarization directions,and of
layer thickness equal to $Lc$ ,we may expect enhancement of SHG in a
crystal quasi-phase-matched for  $d_{33}$  compared with a monodomained crystal
phased matched(PM) for $d_{31}$  of the same length.
The theoretical maximum enhancement factor is found to be
$(d_{33}/d_{31})^{2}(2/\pi)^{2}\approx 23$ ~\cite{FMHW}  .
There have been various attempts to realize quasi-phase-matching. The
method of using
a stack of crystal plates has been applied to GaAs ~\cite{TMA},Quartz, but the
number of plates used
was rather small.Molecular beam epitaxy has been tried with 
GaAs-AlGaAs ~\cite{ DH}
, as well as
growth twins and ferroelectric domains, without precise control of
periods.

   So far the best growth method is developed by Ming et.al.in Nanjing
University, China. Their enhancement factor is about 7 compare to
theoretical 23 ~\cite{FMHW}. In their experiment, the optical superlattice
crystals were grown along the a-axis by the Czochralski technique in a
carefully designed asymmetric temperature field ~\cite{WZF}.
 The starting
materials were prepared by doping some elements like Yi,Er, with a percentile
around $ 5\%$  . After Being b-face cut and polished, the crystals
were etched in 1HF:2HNO3 for ten minutes and examined under an optical
microscope to reveal the ferroelectric domian structures.

Fig.1 shows the optical microscope photograph of a sample ~\cite{WZF}.

\begin{figure}[htb]
\centerline{\psfig{figure=xx.eps,height=2.5in}}

\end{figure}

Fig 2 shows the experiment set up for observing blue and violet light
generation in a
sample of $ Er:LiNbO_{3} $    optical superlattice. When the 808 nm light
injected into the sample, remarkable green fluorescence light and violet 404 nm
second  harmnoic light
were observed even with incident power as low as less than 
200mw ~\cite{LMM}.
\begin{figure}[htb]
\centerline{\psfig{figure=gg.eps,height=2.5in}}
\end{figure}
 

Fig3 shows the harmonic energy dependence on the fundamental energy at
the optimal wavelength of 790nm in a $ Nd:MgO:LiNbO_{3} $   optical
superlattice. It can be seen very clearly that the SHG energy is nearly
proportional to the square of the fundamental energy ~\cite{LLM}.
\begin{figure}[htb]
\centerline{\psfig{figure=zz.eps,height=2.5in}}
\end{figure}
 



\section{Conclusion:}

With the quasiphase matching theory proposed independently by
Bloembergen and
Franken and Ward, the  efficient frequency conversion is achieved in
many nonlinear
optical materials. $LiNbO_{3} $ optical superlattice is among the  excellent
ones. a $LiNbO_{3} $
OSL in bulk form offers the following advantages:direct frequency
doubling of a laser
diode without the need for careful coupling which exists in waveguide
devices, and ease
of phase matching and angle tuning in QPM optical parametric
oscillator(OPO).
The latter is difficult to achieve in waveguide form. Disadvantages: it
is difficult to prepare  a $LiNbO_{3}$  OSL with a small modulating
 period,
which is desirable for SHG to generate short wavelength light using
first-order-QPM;the frequent formation of various island-like domains
during the growth process may destroy the periodicity
of the $LiNbO_{3} $   OSL, and thus  lead to not accurate frequency conversion
and the loss
of SHG energy.

\begin{thebibliography}{50}

\bibitem{ABDP} J.A.Armstrong, N.Bloembergen, J.Ducing, and P.S.Pershan,
Phys.Rev.127,1918(1962)

\bibitem{FMHW} D.Feng,N.B.Ming,J.F.Hong,and
Y.N.Wang,Appl.Phys.Lett.37,607(1980) 

\bibitem{WZF} W.S.Wang,Q.Zhou,and D.Feng,J.Crystal
Growth,79,706(1986) 

\bibitem{FB}G.A.Magel,M.M.Fejer and
R.L.Byer,Appl.Phys.Lett.56,108(1990)

\bibitem{LMCML} Y.L.Lu,L.Mao,S.D.Cheng,N.B.Ming ,and
Y.T.Lu,Appl.Phys.Lett.59,516(1991)

\bibitem{LMM}Y.L.Lu,L.Mao, and N.B.Ming,Appl.Phys.Lett.64,3092(1994)


\bibitem{LLM}Y.L.Lu,Y.Q.Lu,and N.B.Ming,Appl.Phys.Lett.68,2781(1996)

\bibitem{TMA}D.E.Thompson,J.D.McMullen,and
D.B.Anderson,Appl.Phys.Lett.29113(1976) 

\bibitem{DH}C.F.Dewey and
L.O.Hocker,Appl.Phys.Lett.26,442(1975)

\end{thebibliography}

\end{document}





